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Qubit-excitation-based adaptive variational quantum eigensolver

Yordan S. Yordanov, V. Armaos, Crispin H. W. Barnes, David R. M. Arvidsson-Shukur

arXiv:2011.10540v2quant-ph

TL;DR

Molecular VQE needs accurate, optimizable ansätze that remain shallow and hardware-efficient on NISQ devices. The paper introduces QEB-ADAPT-VQE, which grows problem-tailored ansätze from qubit-excitation evolutions, and reports better circuit efficiency and convergence speed than qubit-ADAPT-VQE.

  • Problem

    Molecular VQE requires ansätze that are accurate, easy to optimize, and shallow, while existing approaches face circuit, parameter, or convergence-efficiency trade-offs.

  • Method

    QEB-ADAPT-VQE grows a problem-tailored molecular ansatz iteratively by appending qubit-excitation evolutions, with a modified growth strategy.

  • Results

    QEB-ADAPT-VQE requires asymptotically fewer CNOTs and systematically outperforms qubit-ADAPT-VQE in CNOT efficiency and convergence speed.

  • Takeaways & Limitations

    Qubit-excitation evolutions provide accurate ansatz construction with fewer gates and without as many additional variational parameters and iterations as qubit-ADAPT-VQE.

Abstract

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Molecular simulations with the variational quantum eigensolver (VQE) are a promising application for emerging noisy intermediate-scale quantum computers. Constructing accurate molecular ansätze that are easy to optimize and implemented by shallow quantum circuits is crucial for the successful implementation of such simulations. Ansätze are, generally, constructed as series of fermionic-excitation evolutions. Instead, we demonstrate the usefulness of constructing ansätze with "qubit-excitation evolutions", which, contrary to fermionic excitation evolutions, obey "qubit commutation relations". We show that qubit excitation evolutions, despite the lack of some of the physical features of fermionic excitation evolutions, accurately construct ansätze, while requiring asymptotically fewer gates. Utilizing qubit excitation evolutions, we introduce the qubit-excitation-based adaptive (QEB-ADAPT)-VQE protocol. The QEB-ADAPT-VQE is a modification of the ADAPT-VQE that performs molecular simulations using a problem-tailored ansatz, grown iteratively by appending evolutions of qubit excitation operators. By performing classical numerical simulations for small molecules, we benchmark the QEB-ADAPT-VQE, and compare it against the original fermionic-ADAPT-VQE and the qubit-ADAPT-VQE. In terms of circuit efficiency and convergence speed, we demonstrate that the QEB-ADAPT-VQE outperforms the qubit-ADAPT-VQE, which to our knowledge was the previous most circuit-efficient scalable VQE protocol for molecular simulations.

I. ABSTRACT

The paper addresses the challenge of building molecular VQE ansätze that are accurate, easy to optimize, and shallow enough for NISQ hardware. It introduces QEB-ADAPT-VQE, which uses qubit-excitation evolutions to improve circuit efficiency and convergence while retaining accurate ansatz construction.

  • Motivation: Molecular VQE requires ansätze that accurately approximate ground states, are easy to optimize, and use shallow circuits.These requirements are important because NISQ devices have limited coherence times and VQE trades shallower circuits for more measurements and classical processing.
  • Limitations of existing ansätze: UCC ansätze can be accurate and easy to optimize, but their general-purpose construction introduces redundant parameters, long circuits, and extra overhead for strongly correlated systems.Higher-order excitations and multiple-step Trotterization may be required for strongly correlated systems.
  • Limitations of existing ansätze: Hardware-efficient ansätze provide shallow, flexible circuits but may require many parameters and suffer from vanishing energy gradients because they lack physically motivated structure.The resulting barren-plateau behavior can make classical optimization intractable for large molecules.
  • Iterative VQE: Iterative VQE protocols construct problem-tailored ansätze with optimized parameter counts and circuit depth, but require more quantum measurements.ADAPT-VQE appends parametrized operators selected from an ansatz-element pool using an energy-gradient hierarchy.
  • Qubit-excitation evolutions: Qubit-excitation evolutions combine the accuracy of fermionic excitations with the hardware efficiency of Pauli-string exponentials and use fixed-size circuits.Their circuits act on a fixed number of qubits, whereas fermionic excitation evolutions act on a number scaling at least as O(log2 NMO).
  • QEB-ADAPT-VQE: QEB-ADAPT-VQE grows a problem-tailored molecular ansatz from qubit-excitation evolutions and modifies ansatz growth at the cost of a constant-factor increase in measurements.The protocol was benchmarked through classical numerical simulations of LiH, H6, and BeH2 against UCCSD-VQE, fermionic-ADAPT-VQE, and qubit-ADAPT-VQE.

III. RESULTS

The paper formulates molecular electronic structure as a ground-state eigenvalue problem and maps the molecular Hamiltonian to qubit operators. VQE then minimizes the energy expectation of an ansatz-prepared trial state using alternating quantum measurement and classical parameter updates.

  • Theoretical background and notation: The electronic structure problem seeks the ground-state wavefunction and energy of a molecule by solving the time-independent Schrödinger equation.Under the Born–Oppenheimer approximation, the nuclei are treated as motionless.
  • Theoretical background and notation: The molecular Hamiltonian is expressed in second quantization using fermionic creation and annihilation operators, with one- and two-electron integrals in a spin-orbital basis.The number of Hamiltonian terms scales as O(N^4).
  • Qubit encoding: Jordan–Wigner encoding represents the occupancy of molecular spin-orbital i by the state of qubit i.Fermionic operators are mapped to qubit gate operators with Pauli-z strings accounting for fermionic anticommutation phases.
  • Variational quantum eigensolver: After mapping the Hamiltonian to Pauli operators, VQE minimizes E(θ) = ⟨ψ(θ)|H|ψ(θ)⟩ using the Rayleigh–Ritz variational principle.The quantum computer prepares |ψ(θ)⟩ and evaluates its energy, while a classical computer updates θ iteratively.

B. The ADAPT-VQE protocols

ADAPT-VQE protocols iteratively grow problem-tailored ansätze by selecting unitary ansatz elements from a pool and optimizing their variational parameters. Qubit excitation evolutions offer fixed-size circuits and asymptotically improved CNOT scaling compared with fermionic excitations and Pauli-string constructions.

  • ADAPT-VQE growth: ADAPT-VQE appends unitary ansatz elements iteratively, reoptimizing parameters after each addition to lower the estimated energy.The greedy procedure produces an ansatz tuned to the simulated system and can approximate its ground state with fewer variational parameters than fixed general-purpose ansätze.
  • Protocol pools: Fermionic-ADAPT-VQE samples spin-complement single and double fermionic excitation evolutions, while qubit-ADAPT-VQE uses exponentials of XY-Pauli strings.Both protocols add the pool element with the largest energy-gradient magnitude.
  • Circuit scaling: Fermionic excitation evolutions act on O(NMO) qubits and require circuits with O(NMO) CNOTs, whereas qubit evolutions become increasingly CNOT-efficient for larger systems.The cited fermionic scaling is for Jordan-Wigner encoding; Bravyi-Kitaev scaling is stated as O(log NMO).
  • Qubit excitation elements: Qubit excitation evolutions act on a fixed number of qubits and use fixed-CNOT circuits, with single- and double-excitation implementations requiring 2 and 13 CNOTs.They are represented using combinations of mutually commuting Pauli-string exponentials and permit local circuit optimizations unavailable to Pauli-string exponentials.
  • Circuit and parameter efficiency: Qubit-excitation ansätze reduce the number of variational parameters and can use local circuit optimizations that Pauli-string exponentials do not allow.The comparison assumes the cited CNOT-efficient circuits for qubit and fermionic excitation evolutions and a standard CNOT-staircase construction for Pauli strings.

D. The QEB-ADAPT-VQE protocol

QEB-ADAPT-VQE grows a qubit-excitation ansatz by evaluating energy gradients, testing promising candidates through VQE optimization, and appending the candidate producing the largest energy reduction. Optional spin-complement additions and broader candidate screening can reduce circuit growth and iteration counts.

  • Initialization: QEB-ADAPT-VQE begins with a Hamiltonian gate representation, a pool of unique single and double qubit excitation evolutions, and a reference state with substantial ground-state overlap.The classical simulations use the conventional Hartree-Fock state as the reference.
  • Gradient screening: At each iteration, the protocol prepares the current trial state and measures each pool element’s energy gradient at θp = 0.The gradients are obtained from the expectation value of the commutator between H and the corresponding qubit excitation operator.
  • Candidate selection: The protocol selects n candidates with the largest gradient magnitudes, optimizes each with VQE, and appends the one producing the largest energy reduction.This additional candidate-testing step is absent from the original fermionic-ADAPT-VQE.
  • Candidate-pool size: For n = 10, LiH, H6, and BeH2 achieve a 15% to 25% CNOT reduction at the cost of additional quantum-computer measurements.The n = 1 case corresponds to selecting only the largest-gradient candidate.
  • Termination and cost: The iterative loop stops when the selected energy reduction falls below a positive threshold, otherwise appending the selected evolution and continuing.A worst-case estimate indicates that the protocol may require O(nNMO) quantum-computer measurements.
  • Spin complements: When the ground state is known to have the reference state’s spin, appending spin-complement evolutions with independent parameters roughly halves the iterations needed for a given accuracy.Independent parameters provide flexibility because qubit excitation evolutions do not account for state parity.

E. Classical numerical simulations

The study uses classical numerical VQE simulations of LiH, H6, and BeH2 to compare qubit- and fermionic-excitation ansätze. The calculations use STO-3G spin-orbital bases without frozen orbitals and analytically supplied gradients for BFGS optimization.

  • Molecular test set: Classical numerical simulations compare qubit and fermionic excitations for LiH, H6, and BeH2.H6 is included as a prototype molecule with a strongly correlated ground state.
  • Hamiltonian representation: The molecular Hamiltonians use the STO-3G spin-orbital basis without frozen orbitals, representing LiH, H6, and BeH2 with 12, 12, and 14 qubits.The corresponding numbers of spin-orbitals are also 12, 12, and 14, respectively.
  • Optimization and implementation: Variational parameters are optimized with the gradient-descent BFGS method supplied with analytically calculated energy gradients.The simulations use custom code for arbitrary ansatz-element pools and ansatz-growing strategies.
  • Optimization caveat: At high noise levels, gradient-descent minimizers are likely to struggle to find the global energy minimum, while direct-search minimizers are likely to perform better.This limitation concerns the optimization method used in the simulations rather than the ansatz construction itself.

F. Qubit versus fermionic excitations

The simulations compare qubit- and fermionic-excitation protocols at equilibrium and large bond distances using matched individual-excitation growth. Fermionic ADAPT-VQE converges faster for the more strongly correlated large-distance LiH and BeH2 cases and uses fewer ansatz elements for BeH2.

  • Comparison design: The comparison matches protocols by growing ansätze with individual excitation evolutions, avoiding the parameter-count mismatch from spin-complement pairing.QEB-ADAPT-VQE uses qubit excitations, while fermionic-ADAPT-VQE uses fermionic excitations.
  • Energy convergence: At large bond distances, fermionic-ADAPT-VQE converges faster than QEB-ADAPT-VQE for LiH and BeH2.The difference in convergence rates is reported as larger for these cases than at equilibrium distances.
  • Ansatz compactness: For BeH2, fermionic-ADAPT-VQE requires about 20% fewer ansatz elements on average than QEB-ADAPT-VQE to achieve a given accuracy.The results indicate that fermionic-excitation ansätze can better approximate strongly correlated states than qubit-excitation ansätze.

G. Energy dissociation curves

The QEB-ADAPT-VQE accurately follows molecular energy dissociation behavior, including strongly correlated regimes, while often using fewer variational parameters than UCCSD. Its performance depends on the energy-reduction threshold and molecule.

  • Comparison setup: The comparison uses HF, FCI, VQE, and untrotterized spin-conserving single- and double-excitation UCCSD ansätze in the STO-3G basis.Panels show absolute energies, FCI-referenced errors, and the number of variational parameters required for the target accuracies.
  • Accuracy: UCCSD-VQE achieves chemical accuracy for all LiH bond distances and near-equilibrium H6 and BeH2 distances, but fails farther from equilibrium in H6 and BeH2.These failures occur where the ground states become more strongly correlated.
  • Ansatz size: In LiH and BeH2, QEB-ADAPT-VQE with ϵ6 and ϵ8 is more accurate than UCCSD and uses significantly fewer variational parameters.For H6, QEB-ADAPT-VQE instead requires more parameters on average because its ground state is more strongly correlated.
  • Ansatz size: QEB-ADAPT-VQE parameter counts change abruptly near 1 Å, 2 Å, and 2.75 Å for H6 because molecular structure transformations produce energy-level crossings.Different eigenstates of H become lowest in energy at these crossings.

H. Energy convergence

The energy-convergence study compares QEB-ADAPT-VQE with fermionic- and qubit-ADAPT-VQE using iterations, variational parameters, and CNOT count. QEB-ADAPT-VQE converges faster than qubit-ADAPT-VQE and becomes more CNOT-efficient at higher accuracies.

  • Comparison setup: The protocols are compared on iterations, variational parameters, and CNOTs for LiH, H6, and BeH2 at fixed bond distances and ϵ = 10^-12 Hartree.CNOT count serves as an approximate measure of circuit depth and quantum-subroutine runtime.
  • Iterations: QEB-ADAPT-VQE and fermionic-ADAPT-VQE require similar numbers of iterations when constructing their respective ansätze.This corresponds to approximately equal numbers of qubit- and fermionic-excitation evolutions.
  • Variational parameters: QEB-ADAPT-VQE requires up to twice as many variational parameters as fermionic-ADAPT-VQE.QEB-ADAPT-VQE assigns one parameter per qubit excitation, whereas fermionic-ADAPT-VQE assigns one parameter to spin-complement excitation pairs.
  • Convergence: QEB-ADAPT-VQE systematically requires fewer iterations and variational parameters than qubit-ADAPT-VQE.Single and double qubit excitation evolutions combine 2 and 8 Pauli-string exponentials, respectively.
  • CNOT efficiency: At low accuracies qubit-ADAPT-VQE is more CNOT-efficient, but QEB-ADAPT-VQE systematically outperforms it at higher accuracies.Qubit evolutions permit local circuit optimizations unavailable to Pauli-string evolutions.
  • Fermionic pool comparison: Independent single- and double-fermionic excitation pools converge more slowly but reach higher final accuracies than spin-complement-pair pools.Their greater variational flexibility accounts for the higher achievable accuracy.

IV. DISCUSSION

The paper finds that qubit-excitation evolutions can construct accurate molecular VQE ansätze with fewer CNOTs, motivating QEB-ADAPT-VQE. Simulations indicate faster convergence and better circuit efficiency than qubit-ADAPT-VQE, while leaving important comparison limitations.

  • Qubit-excitation ansätze can approximate molecular electronic wavefunctions almost as accurately as fermionic-excitation ansätze.
  • Qubit-excitation evolutions use fixed-size circuits, whereas fermionic-excitation circuits scale with system size under Jordan-Wigner or Bravyi-Kitaev encoding.
  • Because CNOT count is a primary NISQ cost factor, qubit-excitation evolutions are more suitable for constructing electronic ansätze.
  • QEB-ADAPT-VQE grows a problem-tailored molecular ansatz iteratively by appending single and double qubit-excitation evolutions, and was benchmarked on LiH, H6, and BeH2.
  • QEB-ADAPT-VQE requires up to twice as many variational parameters as fermionic-ADAPT-VQE but asymptotically fewer CNOTs.
  • At low accuracies, qubit-ADAPT-VQE is more CNOT-efficient, but QEB-ADAPT-VQE systematically outperforms it for higher accuracies and larger ansatz circuits.
  • The numerical results are insufficient for a general conclusion, and the comparison ignored a reduced qubit-ADAPT-VQE ansatz-element pool.
  • QEB-ADAPT-VQE converges faster than qubit-ADAPT-VQE, requiring fewer variational parameters and ansatz-construction iterations.

Appendix A: Supplementary note 1

The appendix derives an efficient classical procedure for evaluating trial statevectors and their energy gradients in variational simulations. Storing operator matrices reduces statevector construction cost, while supplying gradients substantially reduces optimization iterations.

  • Statevector construction: The trial state is constructed as a product of exponentials generated by skew-Hermitian qubit-excitation, fermionic-excitation, or Pauli-string operators.The ansatz contains NU elements and starts from a reference state |ψ0⟩.
  • Statevector construction: The resulting trial-state calculation requires NU matrix-to-vector multiplications and NU matrix additions, with total complexity O(NU2^2N_MO).This approach trades computation time for storage of the operator matrices.
  • Gradient evaluation: The energy gradient can be computed with 3NU matrix-to-vector multiplications, giving a cost of O(3NU2^2N_MO).The gradient cost is about three times the trial-state cost.
  • Gradient evaluation: Using the energy-gradient vector reduces VQE iterations by at least an order of magnitude, justifying its use despite the higher per-evaluation cost.The appendix describes gradient-assisted optimization with a BFGS-type minimizer.

Appendix D: Supplementary note 4

This appendix analyzes QEB-ADAPT-VQE complexity in measurements, circuit depth, and runtime. The estimates depend strongly on ansatz size, molecule correlation, optimizer choice, and available parallel quantum devices.

  • Complexity analysis: For an ansatz of m qubit-excitation evolutions, the circuit depth scales as O(m/N_MO).Here m is the QEB-ADAPT-VQE iteration number.
  • Complexity analysis: With BFGS optimization, step 3 requires O(nm^2N_MO^4) quantum measurements in the stated worst-case estimate.The estimate assumes O(m^2) energy evaluations for each of n candidate VQE runs.
  • Complexity analysis: Initializing each new ansatz with the previous parameters plus zero can require fewer evaluations than the worst-case estimate.The worst-case bound assumes all parameters are initialized at zero at every iteration.
  • Complexity analysis: The runtime can be lower bounded by O(m^3/N_MO) for a single parallelized VQE run, because the circuit depth is O(m/N_MO) and optimization uses O(m^2) evaluations.Multiple quantum devices allow the n candidate runs to execute in parallel.
  • Scaling and limitations: The ansatz size NU is problem-specific: strongly correlated states may require O(N_MO^4) qubit-excitation evolutions, while weakly correlated states likely require fewer.Consequently, predicting scaling with N_MO is difficult.
  • Scaling and limitations: Under the stated worst-case assumptions, QEB-ADAPT-VQE has time complexity lower-bounded by O(N_MO^15) and requires O(nN_MO^16) quantum measurements.The comparison passages report UCCSD-VQE bounds of O(N_MO^12) time and O(N_MO^12) measurements.

Appendix E: Supplementary note 5

The appendix studies how the protocol parameter n affects QEB-ADAPT-VQE circuit efficiency for LiH, H6, and BeH2. Increasing n generally reduces CNOT counts, but the benefit is molecule-dependent and does not continue reliably beyond n = 10.

  • Protocol behavior: QEB-ADAPT-VQE greedily selects the qubit-excitation evolution that produces the largest energy reduction when appended to the ansatz.The protocol parameter n controls the number of candidate evolutions considered at each iteration.
  • Simulation setup: The simulations evaluate LiH, H6, and BeH2 ground states in the STO-3G basis at stretched bond distances where correlation effects are stronger.The tested distances are 3 Å for Li–H, H–H, and Be–H bonds.
  • CNOT efficiency: Across all three molecules, n = 30 provides no further CNOT reduction over n = 10 and can yield a slightly smaller reduction.The appendix attributes such inconsistencies to the greedy ansatz-construction strategy and noncommuting qubit evolutions.
  • Trade-offs: Increasing n incurs additional quantum-measurement overhead but is justified when quantum-gate fidelity is the NISQ bottleneck.The authors expect the CNOT-count benefit for n > 1 to increase for larger molecules because their ansatz pools are larger.
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